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An **axiom**, **postulate**, or **assumption** is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (*axíōma*), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.^{[1]}^{[2]}

The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.^{[3]} In modern logic, an axiom is a premise or starting point for reasoning.^{[4]}

In mathematics, an *axiom* may be a "logical axiom" or a "non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (*A* and *B*) implies *A*), while non-logical axioms (e.g., *a* + *b* = *b* + *a*) are substantive assertions about the elements of the domain of a specific mathematical theory, such as arithmetic.

Non-logical axioms may also be called "postulates" or "assumptions". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain.

Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.^{[5]}

**^**Cf. axiom, n., etymology.*Oxford English Dictionary*, accessed 2012-04-28.**^**Stevenson, Angus; Lindberg, Christine A., eds. (2015).*New Oxford American Dictionary*(3rd ed.). Oxford University Press. doi:10.1093/acref/9780195392883.001.0001. ISBN 9780199891535.a statement or proposition that is regarded as being established, accepted, or self-evidently true

**^**"A proposition that commends itself to general acceptance; a well-established or universally conceded principle; a maxim, rule, law" axiom, n., definition 1a.*Oxford English Dictionary*Online, accessed 2012-04-28. Cf. Aristotle,*Posterior Analytics*I.2.72a18-b4.**^**"A proposition (whether true or false)" axiom, n., definition 2.*Oxford English Dictionary*Online, accessed 2012-04-28.**^**See for example Maddy, Penelope (June 1988). "Believing the Axioms, I".*Journal of Symbolic Logic*.**53**(2): 481–511. doi:10.2307/2274520. JSTOR 2274520. for a realist view.